# Summary Information and Formulae MTH109 College Algebra

Size: px
Start display at page:

Transcription

1 Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged) P = Principl (originl deposit, originl lon) r = nnul interest rte (APR) t = time in yers Compound Interest: A = P 1 + r nt n n = compounding periods per yer (number of times interest is figured) Time/Rte/Distnce: d = rt d = distnce trveled r = trvel rte (remember to combine rtes correctly if in moving medium) t = time of trvel Flling Body: s = gt 2 + v 0 t + s 0 s = distnce bove ground (note tht s is positive upwrds, t ground level s = 0) g = ccelertion of grvity ( g 16 ft. or g 4.9 m ) ( sec. ) 2 ( sec. ) 2 t = time flling v 0 = initil velocity s 0 = initil height Formuls for Geometric Objects Squre: A = s 2 nd P = 4s A = re P = perimeter s = one side Rectngle: A = lw nd P = 2l + 2w l = length w = width Circle: A = π r 2 nd C = 2π r C = circumference (perimeter) r = rdius Tringle: A = 1 bh nd P = + b + c 2 b = bse h = height, b, c = sides Right Tringle: 2 + b 2 = c 2, b = legs of the tringle c = hypotenuse Trpezoid: A = 1 2 h ( + b ), b = prllel sides Cube: V = s 3 nd SA = 6s 2 V = volume SA = surfce re Rectngulr Solid: V = lwh nd SA = 2 lw + hl + hw Circulr Cylinder: SA = 2π rh + 2π r 2 V = π r 2 h nd Sphere: V = 4 3 π r3 nd SA = 4π r 2

2 Chpter P: Prerequisites Properties of opertions on the Rel Numbers Closure: Opertion performed on two rel numbers results in rel number. Commuttive: Numbers dd in ny order. Numbers multiply in ny order. Associtive: Addition cn be grouped in ny order. So cn multipliction. Distributive: Multipliction distributes over ddition ( x + y) = x + y Frctions Multiplictive Identity: x y =1 x y = x y = x y Reducing (cnceling): x y = x y =1 x y = x y Adding frctions: d + b d = + b d Multiplying frctions: b x y = x by Simplifying complex frctions: b c d = b d c or b c bd bd d d Absolute Vlues Absolute Vlue: the distnce from zero = if 0 { if < 0 Properties: = 3. b = b 4. b = b = b c bd bd = d bc Distnce AB between points A, B on the number line (coordintes, b): AB = b Properties of Exponents Definition: x n mens x used s fctor (multiplied) n number of times (x is rel number, n is nturl number). Zero power: x 0 =1 Negtive exponent mens reciprocl: x n = 1 x n, 1 x n = x n 1 = x n, If the bses re the sme, multiply: x n x m = x n +m divide: exponentil to power: Power opertes before subtrcts: Power distributes over multipliction: x n x m = x n m ( x n ) m = x n m x y 1 = y x x 2 = ( x x) nd ( x) 2 = ( x) ( x) = x 2 ( xy) n = x n y n

3 Rtionl Exponents nd Rdicls Equivlent forms: m x n = x n m = x n 1m = x 1 m is n nth root of b if: n n = b or = b or = b 1 n Note: Rtionl exponents hve exctly the sme properties s integer exponents Polynomils Monomil in vribles x nd y: x n y m ( is constnt, n nd m non-negtive integers) Coefficient of the monomil: Degree of the monomil: n + m Degree of polynomil: Degree of the highest degree monomil Polynomil in x (generl) n x n + n 1 n n 1 +!+ 1 x + 0 where n 0, degree = n, leding coefficient is n, nd 0 is the constnt term Add (subtrct) polynomils: Combine like terms Multiple polynomils (generl): Multiply every term in first by every term in second. Multiply binomils: FOIL Specil products where u nd v cn be (nd will be) ny lgebric expression: Sum nd Difference u + v Squre of Binomil ( u v) = u 2 v 2 ( u + v) = u 2 + 2uv + v 2 ( u v) = u 2 2uv + v 2 ( u + v) = u 3 + 3u 2 v + 3uv 2 + v 3 ( u v) = u 3 3u 2 v + 3uv 2 v 3 Binomil Cubed Fctoring Formuls: Difference of Squres u 2 v 2 = u + v Perfect Squres u 2 + 2uv + v 2 = u + v Cubes u 3 + v 3 = u + v Grphs Digrm: Stndrd Crtesin Plne Point (x, y) x-coordinte is distnce from the y-xis y-coordinte is distnce from the x-xis Distnce Formul: Distnce between points ( x 1, y 1 ) nd ( x 2, y 2 ): ( u v) 2 u 2 2uv + v 2 = ( u v) 2 u 2 uv + v 2 u 2 + uv + v 2 d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 u 3 v 3 = u v nd Midpoint of line segment joining points x 1, y 1 ( x 2, y 2 ): x Midpoint coordintes: ( x m, y m ) = 1 + x 2, y 1 + y (Note tht this is the verge of x, y coordintes) n Qudrnt II x-xis (negtive) Qudrnt III Origin y-xis (positive) y-xis (negtive). (x, y) Qudrnt I Qudrnt IV x-xis (positive)

4 Chpter 1: Equtions nd Inequlities Tests for symmetry: Grph of n eqution is symmetric with respect to the x-xis Grphicl: If whenever ( x, y) on the grph, ( x, y) is on the grph Algebric: Replcing y with y results in n equivlent eqution y-xis Grphicl: If whenever ( x, y) on the grph, ( x, y) is on the grph Algebric: Replcing x with x results in n equivlent eqution Origin Grphicl: If whenever ( x, y) on the grph, ( x, y) is on the grph Algebric: Replcing both x with x nd y with y results in n equivlent eqution Circle: Set of points ll the sme distnce from center point. Stndrd form: Center is h,k nd rdius is r. ( x h) 2 + y k 2 = r 2 Center t origin: x 2 + y 2 = r 2 Absolute Vlue Equtions u = v where u nd v re lgebric expressions Two possibilities: u = v (vlue of u inside bsolute vlue ws positive) u = v (vlue of u inside bsolute vlue ws negtive) Qudrtic Equtions Zero Product Principle: If b = 0, then = 0 or b = 0 If d > 0 nd x 2 = d Roots: x = ± d. Note there re two roots. Fctored Form: x 2 d = x + d ( x d ) = 0 Qudrtic Eqution - Stndrd Form: x 2 + bx + c = 0 Completing the Squre x + b 2 = b2 4c Qudrtic Formul x = b ± b2 4c 2 Discriminnt b 2 4c > o there re two distinct rel solutions = 0 one rel root, multiplicity of two < 0 two complex solutions, conjugte pir Complex Numbers Define: i = 1, therefore: i 2 = 1 For x, write i x (or x i, but wtch til of rdicl!) for x > 0 Complex number: + bi where nd b re rel numbers nd i = 1 is the rel prt, bi is the imginry prt Imginry number: Any number in the form bi where b is rel number ( = 0)

5 Equlity: + bi = c + di if nd only if = c nd b = d. Addition ( + bi) + ( c + di) = ( + c) + ( b + d)i (dd the rel prts, dd the imginry prts) Subtrction: ( + bi) ( c + di) = ( c) + ( b d)i Multipliction: ( + bi) ( c + di) = ( c bd) + ( d + bc)i (by FOIL nd simplify) Complex conjugte of + bi is bi Product of complex nd its conjugte is rel: ( + bi) ( bi) = 2 + b 2 Division Multiply by 1 using complex conjugte of denomintor: + bi c + di = + bi c + di c di c di = ( c + bd) + ( bc d )i c 2 + d 2 Inequlities - Properties: Trnsitive If < b nd b < c, then < c Addition If < b nd c < d, then + c < b + d Adding constnt If < b, then + c < b + c Multiplying constnt For c > 0 if < b then c < bc For c < 0 if < b then c > bc Solving Inequlities with Absolute Vlue of Liner Algebric Expression u u < : Solve two inequlities for the vrible u < { u < All vlues of u re between - nd. < u <, solve for the vrible u, u > : Solve two inequlities for the vrible u > { u > u < or u >, solve for the vrible Note: open intervl for <, closed intervl for Solving Polynomil Inequlities n x n + n 1 n n 1 +!+ 1 x + 0 < 0 1. Find ll roots of polynomil n x n + n 1 n n 1 +!+ 1 x + 0 = 0 2. The roots re criticl vlues determining intervls to test for solutions 3. Choose representtive vlue to test in ech intervl between criticl vlues 4. If test vlue mkes inequlity true, then ll vlues in tht intervl re solutions. If test vlue is flse, ll vlues in tht intervl between criticl vlues re not solutions.

6 Chpter 2: Functions nd Their Grphs SOLUTION to n eqution (or other sttement), the roots of n eqution (function), the zeros of n eqution (function): ny nd ll vlues tht will mke the eqution TRUE. GRAPH: picture of ll solutions to sttement. Solutions re plotted on n pproprite set of xes. Function: Function f from set A to set B, is reltion which ssigns one x in set A, the domin, to exctly one y in set B, the rnge. 1. Ech element in A must be mtched to n element in B. 2. Not ll elements in B my be mtched with n element in A. 3. Two or more elements in A my be mtched to n element in B. 4. No element in A is mtched to more thn one element in B. Verticl Line Test: A grph represents function if no verticl line intersects the grph t more thn one point (no points is oky). Imge: Element y (dependent vrible) is the imge of x (independent vrible) Domin: set of ll inputs to function, set of ll independent vribles, set of from vlues in mpping. The x vlues. Rnge: set of ll outputs from function, set of ll dependent vribles, set of to vlues in mpping. This is the set of ll imges of the domin. The y vlues. Common limits on Domin: Division by zero (expression in denomintor cnnot be zero) Even roots of negtive vlues of the rdicnd (for rel numbers) Implied domin - function is not defined there. (e.g. rectngle width less thn zero) Liner Equtions Generl Form: Ax + By + C = 0 Slope-Intercept Form: y = mx + b where the slope is m nd the y-intercept is the point 0,b Point-Slope Form: y y 1 = m( x x 1 ) Two-Point Form: y y 1 = y 2 y 1 ( x x 1 ) x 2 x 1 Verticl Line: x = Horizontl Line: y = b Slope: "... is rise over run." Chnge in the y coordinte for given chnge in the x coordinte m = y 2 y 1 x 2 x 1 Prllel Lines: Two distinct, non-verticl lines re prllel if nd only if their slopes re equl. m 1 = m 2 Perpendiculr Lines: Two distinct, non-verticl lines re perpendiculr if nd only if their slopes re negtive reciprocls of ech other. (The product of slopes is -1.) m 2 = 1 m 1 or m 1 m 2 = 1 Function nme Input vlue Domin element f(x) = y Output vlue Rnge element

7 Grphs of Functions Zeros of function: All vlues of x where f ( x) = 0 Incresing Function: x 1 < x 2 implies f ( x 1 ) < f ( x 2 ) for the given intervl Decresing Function: x 1 < x 2 implies f ( x 1 ) > f ( x 2 ) for the given intervl Constnt Function: f ( x 1 ) = f ( x 2 ) for ny x 1 nd x 2 in the given intervl Reltive Minimum: Vlue in n intervl ( x 1, x 2 ) where f ll other x ( x 1, x 2 ). Reltive minimum is point (, f ( ) ) Reltive Mximum: Vlue in n intervl ( x 1, x 2 ) where f ll other x ( x 1, x 2 ). Reltive minimum is point (, f ( ) ) Averge Rte of Chnge of Function is the slope of the secnt line between the two function points: Chnge in f x < f ( x) for > f ( x) for for given chnge in the x, or m sec = f ( x 2 ) f ( x 1 ) = f ( x) (Like x tken to even power.) (Like x tken to odd power.) Even Function: f x Odd Function: f ( x) = f x Liner Function: f ( x) = mx + b Slope = m, y-intercept ( f ( x) = 0) = b or the point (0, b) x-intercept = f ( 0) = b m or the point b m, 0 Grph is incresing for m > 0, decresing for m < 0 Squring Function: f ( x) = x 2 Incresing on ( 0, ) Even function Domin: set of ll rel numbers. Rnge: 0, Odd function [ ) Grph: Intercept t 0,0 Decresing on (,0) Incresing on ( 0, ) Symmetric with respect to y-xis Reltive minimum t 0,0 = x 3 Cubic Function: f x Odd function Domin: set of ll rel numbers. Rnge: set of ll rel numbers. Grph: Intercept t 0,0 Incresing on, Symmetric with respect to origin Squre Root Function: f x [ ) [ ) Domin: 0, Rnge: 0, Grph: Intercept t 0,0 = x Reciprocl Function: f ( x) = 1 x x 2 x 1 Domin: (,0) ( 0, ), or x 0 Rnge: (,0) ( 0, ), or f x Grph: No intercepts Decresing on (,0) nd ( 0, ) Symmetric with respect to origin Absolute Vlue: f ( x) = x Even function Domin: set of ll rel numbers. Rnge: 0, [ ) Grph: Intercept t 0,0 Decresing on (,0) Incresing on ( 0, ) Symmetric with respect to y-xis Reltive minimum t ( 0,0) Gretest Integer: f x integer less thn or equl to x 0 = [[x]] = gretest

8 Trnsformtions of Grphs Grph of given function y = f ( x), nd where c is positive rel number Verticl nd Horizontl Shifts of grph Verticl shift c units up: h( x) = f ( x) + c Verticl shift c units down: h( x) = f ( x) c Horizontl shift c units right: h( x) = f ( x c) Horizontl shift c units left: h( x) = f ( x + c) Reflections of grphs Reflection in x-xis: h( x) = f ( x) Reflection in y-xis: h( x) = f ( x) Nonrigid Trnsformtions Verticl stretch h( x) = c f ( x) where c > 1 Verticl shrink h( x) = c f ( x) where 0 < c < 1 Horizontl stretch h( x) = f ( c x) where 0 < c < 1 Horizontl shrink h( x) = f ( c x) where c > 1 Arithmetic of Functions Functions f nd g hve overlpping domins Sum: ( f + g) ( x) = f ( x) + g( x) Difference: ( f g) ( x) = f ( x) g( x) Product: ( fg) ( x) = f ( x) g( x) Quotient: f ( x) = f ( x ) g g( x) ( x ) 0 Composition of Functions Composition of f with g ( f! g) ( x) = f ( g( x) ) Domin of ( f! g) All x in domin of g such tht g x domin of f. Inverse Functions Nottion: Function is f, Inverse of function f is written f 1 nd red "f inverse" is in the Grph: f 1 is the reflection of grph of f cross the line y = x. Not ll functions hve inverses Horizontl Line Test: f hs n inverse if nd only if no horizontl line intersects the grph of f t more thn one point. One-to-One Functions: Ech vlue of the dependent vrible corresponds to exctly one vlue of the independent vrible. It is function (verticl line test), nd in reverse - ny one vlue of f ( x) is given by only one x (horizontl line test). One-to-one functions hve inverses. Finding the Inverse of Function 1. Verify tht f ( x) is one-to-one. 2. Use y in plce of f ( x). 3. Swp vribles in the eqution: Put y where x ws nd x where y ws. 4. Solve for y. 5. The y is now f 1 ( x)

9 Chpter 3: Polynomil functions Polynomil function with degree n: f ( x) = n x n + n 1 n n 1 +!+ 1 x + 0 n is nonnegtive integer, nd n, n 1,!, re rel numbers with n 0 = x 2 + bx + c Qudrtic Function: Polynomil function with degree 2 or f x Grph prbol Grph symmetric to xis of symmetry (or just xis) Axis intersects prbol t the vertex > 0, prbol opens up, vertex is minimum t x = b 2 < 0, prbol opens down, vertex mximum t x = b 2 Vertex Point b 2, f b 2 Stndrd Form of Qudrtic Functions Polynomil Function f ( x) = ( x h) 2 + k Axis is verticl line: x = h Vertex is point ( h,k) = n x n + n 1 n n 1 +!+ 1 x + 0 hs degree n f x If n is odd, left nd right tils of grph go in opposite directions: n > 0 left til to flls without bound, right til rises without bound. n < 0 left til to rises without bound, right til flls without bound. There must be t lest one rel zero (root). Grph must cross x-xis. If n is even, left nd right tils of grph go in the sme directions: n > 0 left til to rises without bound, right til rises without bound. n < 0 left til to flls without bound, right til flls without bound. Turning points: Grph hs t most n-1 turning points where the grph chnges from incresing to decresing or vice vers. These turning points re lso reltive mxim or reltive minim. Zeros: f ( x) hs t most n rel zeros. Rel Zeros of polynomil function f ( x): These ll men the sme thing Zero: A rel number vlue such tht f ( ) = 0 Solution: x = is solution if f ( x) = 0 Fctor: Quntity ( x ) is fctor of the polynomil f ( x) x-intercept Point,0 Repeted Zeros is n x-intercept of f ( x) hs binomil fctors ( x ) k, for k > 1, then f ( x) hs If polynomil function f x repeted zero, x = of multiplicity k. If k is odd, the grph crosses the x-xis t x =. If k is even, the grph touches (but does not cross) the x-xis t x =.

10 Intermedite Vlue Theorem Let nd b be rel numbers such tht < b nd f is polynomil function with f ( ) f ( b). In the intervl [,b], f tkes on every vlue between f ( ) nd f ( b). Note especilly if there is sign chnge between f ( ) nd f ( b), f pssed through zero nd there is root in [,b]. Division of polynomils Dividend = f ( x), Divisor = d( x), Quotient = q( x), ) d(x) Reminder r( x) (nd wtch your like terms!) f ( x) = d( x) q( x) + r( x) Synthetic Division of x 3 + bx 2 + cx + d by ( x h): h 1+ b c d 2x h h b c d j k R Quotient: q x Synthetic division by h: 3+ h b c d h 4x = x 2 + jx + k Reminder: r( x) = R Reminder is vlue of polynomil t h If reminder is 0, ( x h) is fctor of the polynomil If reminder is 0, h is root nd ( h,0) is the x-intercept Fundmentl Theorem of Algebr Every polynomil f(x) of degree n > 0 hs exctly n zeros (roots) mong the complex numbers (including multiplicity). Liner Fctoriztion Theorem: Every polynomil f(x) of degree n > 0 hs exctly n liner fctors. = n ( x c 1 )( x c 2 )! ( x c n ) f x where c 1, c 2,!, c n re complex numbers Reminder Theorem: After synthetic division by h, R = f ( h), i.e. the reminder is the vlue of the polynomil t x = h. Doing synthetic division by h is often esier thn evluting f ( h) for higher degree polynomils. Fctor Theorem: Polynomil f(x) hs fctor ( x h) iff f ( h) = 0... nd h is root, solution, zero, etc. of the polynomil f(x). Polynomil f(x) hs zero t x = h iff x h Rtionl Zero Theorem: for f x coefficients re integers: is fctor of f(x) = n x n + n 1 n n 1 +!+ 1 x + 0 where ll The only possible rtionl roots will hve the form p q where p is ± ll fctors of 0, the constnt term q is ± ll fctors of n, the leding coefficient Conjugte Zeros Theorem: If polynomil hs complex root, + bi, the complex conjugte, bi is lso root., i.e. complex roots pper in pirs. etc. q(x) f(x) r(x)

11 Chpter 4: Rtionl Functions nd Conics [Conics omitted] Intercepts - Generl y-intercept(s): evlute f ( 0) =? x-intercept(s): set f ( x) = 0, solve for x =? Asymptote: stright line which curve of grph cn pproch but never touch. 1. The line x = is verticl symptote of the grph of f if f ( x) or f ( x) s x from the right or from the left. 2. The line y = b is horizontl symptote of the grph of f if f x b s x or x. = N ( x ) D( x) Rtionl function: f x Where N ( x) nd D( x) hve no common fctors 1. Hs verticl symptote t ll roots (zeros) of D( x). 2. Hs one or no horizontl symptote depending on the degrees of N ( x) nd D( x): If the degree of N ( x) < degree of D( x), then f hs horizontl symptote on the line y = 0, the x-xis. If the degree of N ( x) = degree of D( x), then f hs horizontl symptote on the line y = n b m (y = rtio of leding coefficients). 3. If the degree of N ( x) is exctly one lrger thn the degree of D( x), then f hs slnt (oblique) symptote long the line determined by q( x), the first degree polynomil found by diving the numertor by the denomintor: N ( x) D( x) = q( x) + r( x) 4. If there is common fctor ( x h), between the numertor nd denomintor of rtionl function, the grph hs hole in where x = h. Horizontl Asymptotes of Rtionl Functions Using Limit Ide Rtionl function is rtio of polynomils: R( x) = n x n + n 1 x n 1 + n 2 x n b m x m + b m 1 x m 1 + b m 2 x m Degree of numertor = n nd degree of denomintor = m. 1. Fctor x n from ech term in numertor nd x m from ech term in denomintor 2. Cncel x n with x m. 3. Tke the limit s x goes to infinity of the tht expression. 4. Result is the eqution of the symptote: Horizontl Asymptotes of Rtionl Functions If n - m > 1, there is no symptote. If n - m = 1, the eqution will be liner eqution in x n oblique symptote. If n = m, it will be the horizontl line y = n b m If n < m, it will be the x xis, the line y = 0.

12 Chpter 5: Exponentil nd Logrithmic Functions Exponentil function: f ( x) = x ( > 0, 1, x is rel) Note: Exponent is vrible x Note: When bse 0 < < 1, n equivlent form exists with bse 1 (which is greter thn 1) nd n exponent of -x. Therefore, we usully use bse > 1. Properties: f ( x) = x For > 1: Properties: f ( x) = x For > 1: Horizontl symptote to y-xis on left o Horizontl symptote to y-xis on right Hs y-intercept ( 0,1 ) Hs y-intercept ( 0,1 ) Is strictly incresing Is strictly decresing Generl Properties: Domin is ll rel numbers. Rnge is y > 0. Function is one-to-one If u = v, then u = v If u = b u, (, b > 0), then either u = 0 or = b If u = v then u = v If u = b u (where, b > 0) then either u = 0 or = b. Nturl Exponent Function f ( x) = e x, where e = lim m or e m m Models using the Exponentil Functions: Compound interest (specified compounding periods) formul: A = P 1 + r nt n A = Amount vilble P = Amount invested r = nnul interest rte n = number of times compounded nnully t = time in yers Compound interest with continuous compounding: A = Pe rt Exponentil growth Q( t) = q 0 e kt q 0 is quntity t t = 0 k is the growth constnt Exponentil decy Q t = q 0 e kt Logrithms Inverse of exponentil: If f x Equivlence: log x = y y = x Inverse of exponentil Common Logrithm: Nturl Logrithm: Logrithmic Identities: log 1 = 0 If f x = x, then f 1 ( x) = log x = x, then f 1 ( x) = log x log x = log 10 x = n where n is 10 n = x ln x = log e x = n where n is e n = x log x = x log =1 log x = x If log u = log v then u = v If log u = log b u, then u = 1, or = b.

13 Properties of logrithms: 1. log xy = log x + log y 2. log x y = log x log y 3. log x n = nlog x Chnge of bse: log b x = log x log b Solving Exponentil/Logrithmic Equtions 1. Use one-to-one property: If u = v, then u = v If log u = log v then u = v 2. Use equivlent forms: log x = y y = x 3. Use inverse opertions: = x, then f 1 ( x) = log x nd log x = x = log x, then g 1 ( x) = x nd log x = x f x g x Chpter 9: Systems of Equtions Solve by grphing: Grph equtions. Find intersection point for pproximte solution(s). Solve by Substitution: Solve one eqution for one vrible in terms of the other Substitute solved expression for first vrible Solve resulting eqution in one vrible Bck substitute to find the other vrible Solve by Elimintion: Multiply one or both equtions to obtin one pir of coefficients which differ only in sign Add the equtions to eliminte tht vrible nd solve the resulting eqution in the other vrible Bck substitute to find the first vrible Distnce = Trvel Rte Time Brek-even Point: Revenues = Costs Equilibrium Point: Supply = Demnd System of two liner equtions in two vribles l 1 : y = m 1 x + b 1 nd l 2 : y = m 2 x + b 2 Consistent System when m 1 m 2 : Exctly one solution Grphs of l 1 nd l 2 intersect t exctly one point Inconsistent Systems when m 1 = m 2 : Infinitely mny solutions if lines coincide (re identicl) No solutions if lines re prllel nd do not coincide Gussin Elimintion: System of liner equtions is mnipulted until it is in tringulr form where the only non-zero coefficient for the first vrible occurs in the first row, the only non-zero coefficients for the second vrible occur in the first nd second equtions, nd so on. The system cn now be solved by bck substitution.

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

### Precalculus Spring 2017

Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

### CHAPTER 9. Rational Numbers, Real Numbers, and Algebra

CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers

### Chapter 1: Fundamentals

Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

### Identify graphs of linear inequalities on a number line.

COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

### SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

### The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+

.1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### MATH 144: Business Calculus Final Review

MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

### 1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

### The graphs of Rational Functions

Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior

### REVIEW Chapter 1 The Real Number System

Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

### TO: Next Year s AP Calculus Students

TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

### AB Calculus Review Sheet

AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

### Math Sequences and Series RETest Worksheet. Short Answer

Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning

### MATH SS124 Sec 39 Concepts summary with examples

This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

### ( ) as a fraction. Determine location of the highest

AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

### ( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### Unit 1 Exponentials and Logarithms

HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

### Adding and Subtracting Rational Expressions

6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy

### I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics

AAT-A Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #17-45 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped

### ( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

### Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

### Integral points on the rational curve

Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin - Mrinette 750 W. Byshore Street Mrinette, WI 5443-453 Also: Konstntine Zeltor P.O. Box

### fractions Let s Learn to

5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

### ( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht

### A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

### Topics for final

Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit

### THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

### 1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER

### I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

### Chapter 1 Cumulative Review

1 Chpter 1 Cumultive Review (Chpter 1) 1. Simplify 7 1 1. Evlute (0.7). 1. (Prerequisite Skill) (Prerequisite Skill). For Questions nd 4, find the vlue of ech expression.. 4 6 1 4. 19 [(6 4) 7 ] (Lesson

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### B Veitch. Calculus I Study Guide

Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

### P 1 (x 1, y 1 ) is given by,.

MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

### Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor

### Review of basic calculus

Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

### Advanced Algebra & Trigonometry Midterm Review Packet

Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

### Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

### REVIEW SHEET FOR PRE-CALCULUS MIDTERM

. If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,

### Main topics for the Second Midterm

Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### Lesson 2.4 Exercises, pages

Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0 - ( ) () 0 c) -7 + d) (7) ( ) 7 - + 8 () ( 8). Expnd nd simplify. ) b) - 7 - + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing

### Math 154B Elementary Algebra-2 nd Half Spring 2015

Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know

### Obj: SWBAT Recall the many important types and properties of functions

Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions One-to-One nd Inverse Functions

### How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

### APPLICATIONS OF THE DEFINITE INTEGRAL

APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through

### Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

### Sample Problems for the Final of Math 121, Fall, 2005

Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

### Mathematics for economists

Mthemtics for economists Peter Jochumzen September 26, 2016 Contents 1 Logic 3 2 Set theory 4 3 Rel number system: Axioms 4 4 Rel number system: Definitions 5 5 Rel numbers: Results 5 6 Rel numbers: Powers

### Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

### 1.) King invests \$11000 in an account that pays 3.5% interest compounded continuously.

DAY 1 Chpter 4 Exponentil nd Logrithmic Functions 4.3 Grphs of Logrithmic Functions Converting between exponentil nd logrithmic functions Common nd nturl logs The number e Chnging bses 4.4 Properties of

### Bridging the gap: GCSE AS Level

Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

### Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.

Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it

### UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

### Math 3B Final Review

Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:45-10:45m SH 1607 Mth Lb Hours: TR 1-2pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### Pre-Calculus TMTA Test 2018

. For the function f ( x) ( x )( x )( x 4) find the verge rte of chnge from x to x. ) 70 4 8.4 8.4 4 7 logb 8. If logb.07, logb 4.96, nd logb.60, then ).08..867.9.48. For, ) sec (sin ) is equivlent to

### Integration Techniques

Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

### BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

### List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

### MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

### Math 1B, lecture 4: Error bounds for numerical methods

Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

### First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009

Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No

### AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

### Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must

### APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

### 15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

- TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

### Chapters Five Notes SN AA U1C5

Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

### Chapter 1: Logarithmic functions and indices

Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

### Math 113 Exam 1-Review

Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

### Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.

Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd

### Math 116 Calculus II

Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### The use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.

ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion

### 3.1 Exponential Functions and Their Graphs

. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

### Faith Scholarship Service Friendship

Immcult Mthemtics Summer Assignment The purpose of summer ssignment is to help you keep previously lerned fcts fresh in your mind for use in your net course. Ecessive time spent reviewing t the beginning

### Mathematics Extension 1

04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### MONROE COMMUNITY COLLEGE ROCHESTER, NY MTH 104 INTERMEDIATE ALGEBRA DETAILED SOLUTIONS. 4 th edition SPRING 2010

MONROE COMMUNITY COLLEGE ROCHESTER, NY * MTH 04 INTERMEDIATE ALGEBRA FINAL EXAM REVIEW DETAILED SOLUTIONS * 4 th edition SPRING 00 Solutions Prepred by Prof. Jn (Yhn) Wirnowski Evlute the function in problems:

### Lecture 0. MATH REVIEW for ECE : LINEAR CIRCUIT ANALYSIS II

Lecture 0 MATH REVIEW for ECE 000 : LINEAR CIRCUIT ANALYSIS II Aung Kyi Sn Grdute Lecturer School of Electricl nd Computer Engineering Purdue University Summer 014 Lecture 0 : Mth Review Lecture 0 is intended

### NAME: MR. WAIN FUNCTIONS

NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors